Here, we are interested in variational quantum algorithms for solving differential equations [1]}, such as the Black-Scholes equation [2]}, [3]}, the Poisson equation [4]}, [5]}, and the Helmholtz equation [6]}. Specifically, the Poisson equation can be solved efficiently through explicit decomposition of the coefficient matrix derived from finite difference discretization [4]} using minimal cost function evaluations [5]} and shallower circuit depth compared to other non-variational quantum algorithms [9]}, [10]}, [11]}, [1]}. A natural question to ask, then, is whether such variational algorithms for Poisson equations can be extended to solving evolution equations, i.e. partial differential equations including a time domain. McArdle et al. [13]} proposed a variational quantum algorithm which simulates the real (imaginary) time evolution of parametrized trial states via forward Euler time-stepping of the Wick rotated Schrödinger equation, thereby solving the Black-Scholes equation, and by extension, the heat equation [2]}, [3]}. Besides issues of ansatz selection and quantum complexity, time-stepping based on an explicit Euler method may be unstable, a limiting condition exacerbated by noise.
